Quantitative stability of the free boundary in the obstacle problem

Serfaty, Sylvia; Serra, Joaquim

doi:10.2140/apde.2018.11.1803

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Quantitative stability of the free boundary in the obstacle problem

Sylvia Serfaty and Joaquim Serra

Vol. 11 (2018), No. 7, 1803–1839

DOI: 10.2140/apde.2018.11.1803

Abstract

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in $ℝ^{n}$ ( $n \geq 2$ ) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field.

To do so, working in the setting of the whole space, we examine the evolution of the free boundary $Γ^{t}$ corresponding to the boundary of the contact set for a family of obstacle functions $h^{t}$ . Assuming that $h = h^{t} (x) = h (t, x)$ is $C^{k + 1, α}$ in $[- 1, 1] \times ℝ^{n}$ and that the initial free boundary $Γ^{0}$ is regular, we prove that $Γ^{t}$ is twice differentiable in $t$ in a small neighborhood of $t = 0$ . Moreover, we show that the “normal velocity” and the “normal acceleration” of $Γ^{t}$ are respectively $C^{k - 1, α}$ and $C^{k - 2, α}$ scalar fields on $Γ^{t}$ . This is accomplished by deriving equations for this velocity and acceleration and studying the regularity of their solutions via single- and double-layer estimates from potential theory.

Keywords

obstacle problem, contact set, coincidence set, stability, equilibrium measure, potential theory

Mathematical Subject Classification 2010

Primary: 35R35, 31B35, 49K99

Milestones

Received: 10 August 2017

Revised: 13 February 2018

Accepted: 9 April 2018

Published: 20 May 2018

Authors

Sylvia Serfaty
Courant Institute of Mathematical Sciences, New York University New York, NY United States

Joaquim Serra
Department of Mathematics ETH Zürich Zürich Switzerland

Vol. 11, No. 7, 2018